Descent for fibred monads
Dear categorists, Does the following variant of the Benabou-Roubaud/Beck monadic descent theorem appear anywhere? Let p:E--->B be a fibration with sums and let T:E--->E be a fibred monad over B. Let q: E^T ----> B be the induced fibration of T-algebras. Let f: x--->y in B. Then to give T-algebra descent data for f---that is, a diagram over the kernel-pair of f valued in E^T---is equally to give an algebra for the composite monad E_x ----f_!----> E_y ----T_y---> E_y ---f^*----> E_x This doesn't seem to be an application of the usual monadic descent theorem to q: E^T ---> B; that would identify T-algebra descent data for f with algebras for a monad on (E^T)_x, not on E_x. For example, take E ----> S a connected topos with pi_0 -| Delta -| Gamma. Let T be the monad for constant objects on E induced by the fibred adjunction pi_0 -| Delta. Given f: U --->> 1 in E, to give T-algebra descent data for f is to give a locally constant object split by U. So such objects are equally the algebras for the monad E/U -----> E/U (A--->U) |----> (Delta pi_0 A) x U ----> U In the same situation, take T to be the monad for free vector spaces E ---pi_0---> S ---Fv---> S ---Delta---> E induced by the free vector space monad Fv on S. Then T-algebra descent data over U --->> 1 is a vector bundle split by U; so such objects are equally algebras for the monad (A--->U) |----> (Delta Fv pi_0 A) x U ---> U Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Actually
In the same situation, take T to be the monad for free vector spaces E ---pi_0---> S ---Fv---> S ---Delta---> E induced by the free vector space monad Fv on S. Then T-algebra descent data over U --->> 1 is a vector bundle split by U; so such objects are equally algebras for the monad (A--->U) |----> (Delta Fv pi_0 A) x U ---> U
This bit is clear rubbish. But the rest of my question remains. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Richard, I would like to see your question formulated more precisely, and showing general (admissible) Galois theory example instead of the locally connected topos example. Some days ago you recommended Carboni-Janelidze-Kelly-Pare paper as one of references for factorization systems (thank you for that!), and now please look at Section 5 of that paper. Note that "admissible"="semi-left-exact" can be replaced with "fibration". Best regards, George Janelidze -------------------------------------------------- From: "Richard Garner" <richard.garner@mq.edu.au> Sent: Thursday, May 15, 2014 1:15 PM To: "Categories list" <categories@mta.ca> Subject: categories: Descent for fibred monads
Dear categorists,
Does the following variant of the Benabou-Roubaud/Beck monadic descent theorem appear anywhere?
Let p:E--->B be a fibration with sums and let T:E--->E be a fibred monad over B. Let q: E^T ----> B be the induced fibration of T-algebras. Let f: x--->y in B. Then to give T-algebra descent data for f---that is, a diagram over the kernel-pair of f valued in E^T---is equally to give an algebra for the composite monad
E_x ----f_!----> E_y ----T_y---> E_y ---f^*----> E_x
This doesn't seem to be an application of the usual monadic descent theorem to q: E^T ---> B; that would identify T-algebra descent data for f with algebras for a monad on (E^T)_x, not on E_x.
For example, take E ----> S a connected topos with pi_0 -| Delta -| Gamma. Let T be the monad for constant objects on E induced by the fibred adjunction pi_0 -| Delta. Given f: U --->> 1 in E, to give T-algebra descent data for f is to give a locally constant object split by U. So such objects are equally the algebras for the monad
E/U -----> E/U (A--->U) |----> (Delta pi_0 A) x U ----> U
In the same situation, take T to be the monad for free vector spaces E ---pi_0---> S ---Fv---> S ---Delta---> E induced by the free vector space monad Fv on S. Then T-algebra descent data over U --->> 1 is a vector bundle split by U; so such objects are equally algebras for the monad (A--->U) |----> (Delta Fv pi_0 A) x U ---> U
Richard
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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George Janelidze -
Richard Garner